Why would my car make a grinding noise when i start it and after running for a bit the grinding noise becomes slower?

Don't ignore that kind of sound. Find out what is causing it and replace or repair the faulty part.

A "grinding noise" can be caused by an alternator, water pump, idler pulley, heater fan motor, air pump or any of a long list of components, many of which can cause some serious problems if they seize or come apart.